May the Merciful One bless (my father) the head of this house and (may mother) the lady of the house...

Why do we first give a בְּרָכָה (blessing) to the בַּיִת (house) and the שֻׁלְחָן (table), then ask for אֵלִיָּהוּ הַנָּבִיא (Elijah the Prophet) and then go back and give a בְּרָכָה (blessing) to the בַּעַל וּבַּעֲלַת הַבַּיִת (heads of the house)?

2 Answers
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In one of his letters, R. Yosef Yitzchak Schneersohn (the sixth Lubavitcher Rebbe) explains that the order of the "harachaman"s follows the order of the ten sefiros, from the lowest (malchus) to the highest (keser).

Specifically, then, these three are in this order, going from "left" to "right":

The previous one, ...הרחמן הוא ישבור עול, is associated with gevurah (strictness, severity), with with Hashem breaks (ישבור) His enemies.

...הרחמן הוא ישלח ברכה is associated with tif'eres ("beauty," a blend of chesed and gevurah), through which Hashem's blessings flow down to us.

(He then notes that his father and predecessor, R. Shalom DovBer Schneersohn, once explained it the other way around, going from right to left: הרחמן הוא ישבור is chesed, הרחמן הוא ישלח ברכה is tif'eres, and הרחמן הוא ישלח לנו את אליהו is gevurah.)

Next are אבי מורי and אמי מורתי, representing the "cerebral" sefiros of chochmah (wisdom) and binah (understanding); for this reason he says (and this indeed is the Chabad custom) that these phrases are said even when one is not eating at his or her parents' table, and even after they have passed away.

The bracha for the baal habayis is a formal bracha made at the culmination of the other brachos (it is hinted to by the word "es" in the pasuk). Initially it came after the 3 brachos of benching. Later, after the fourth bracha. The benching was further extended by the harachamans, after which came the bracha for the baal habayis (which morphed into a harachaman itself) and that was the end of benching- "V'NOMAR AMEN!" The rest of benching was either added later, or not considered a significant part of the benching.